How many digits?

How many digits does the number 2^100 have? Using complicated caculator softwares it is 31 digits but how will you find out with pencil and paper

Hello, sundance240!

Quote:

---

How many digits does the number 2^100 have?
Using complicated caculator softwares, it has 31 digits,
but how will you find out with pencil and paper?

---

We note that: .2^{10} .= .1024 .≈ .10^3

Hence: .2^{100} .≈ .(10^3)^{10} .= .10^{30}

. . which has 31 digits.

Quote:

---

Originally Posted by sundance240

How many digits does the number 2^100 have? Using complicated caculator softwares it is 31 digits but how will you find out with pencil and paper

---

In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... . Therefore log(2^{100}) = 100*log(2) = 30.10..., and since this is between 30 and 31 it follows that 2^{100} is a 31-digit number.

Quote:

---

Originally Posted by Opalg

In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... .

---

Wow...

Quote:

---

Originally Posted by Opalg

In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... . Therefore log(2^{100}) = 100*log(2) = 30.10..., and since this is between 30 and 31 it follows that 2^{100} is a 31-digit number.

---

Wow! What a dinosaur. I suppose Napier's bones are fossils by now? (Rofl)

-Dan

Quote:

---

Originally Posted by Opalg

In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... . Therefore log(2^{100}) = 100*log(2) = 30.10..., and since this is between 30 and 31 it follows that 2^{100} is a 31-digit number.

---

of course, the "log" button on any reputable brand of calculator (including the generic one shipped with any version of Windows) will accomplish the same thing.

Quote:

---

Originally Posted by topsquark

Wow! What a dinosaur. I suppose Napier's bones are fossils by now? (Rofl)

-Dan

---

Years "B.C."- before calculators.